- Research Article
- Open Access

# Sharpening the Becker-Stark Inequalities

- Ling Zhu
^{1}Email author and - Jiukun Hua
^{1}

**2010**:931275

https://doi.org/10.1155/2010/931275

© L. Zhu and J. Hua 2010

**Received:**3 April 2009**Accepted:**14 January 2010**Published:**20 January 2010

## Abstract

In this paper, we establish a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one.

## Keywords

- Power Series
- Natural Number
- Series Expansion
- Zeta Function
- Power Series Expansion

## 1. Introduction

Steckin [1] (or see Mitrinovic [2, 3.4.19, page 246]) gives us a result as follows.

Theorem 1.1 (see [1, Lemma ]).

Later, Becker and Stark [3] (or see Kuang [4, 5.1.102, page 248]) obtain the following two-sided rational approximation for .

Theorem 1.2.

Furthermore, and are the best constants in (1.2).

In fact, we can obtain the following further results.

Theorem 1.3.

Furthermore, and are the best constants in (1.3).

In this paper, in the form of (1.2) and (1.3) we shall show a general refinement of the Becker-Stark inequalities as follows.

Theorem 1.4.

where are the even-indexed Bernoulli numbers.

Furthermore, and are the best constants in (1.4).

## 2. Four Lemmas

## 3. Proofs of Theorems

## Authors’ Affiliations

## References

- Steckin SB: Some remarks on trigonometric polynomials.
*Uspekhi Matematicheskikh Nauk*1955, 10(1(63)):159–166.MathSciNetGoogle Scholar - Mitrinovic DS:
*Analytic Inequalities*. Springer, New York, NY, USA; 1970:xii+400.View ArticleMATHGoogle Scholar - Becker M, Strak EL: On a hierarchy of quolynomial inequalities for tanx.
*University of Beograd Publikacije Elektrotehnicki Fakultet. Serija Matematika i fizika*1978, (602–633):133–138.Google Scholar - Kuang JC:
*Applied Inequalities*. 3rd edition. Shangdong Science and Technology Press, Jinan City, China; 2004.Google Scholar - Scharlau W, Opolka H:
*From Fermat to Minkowski, Undergraduate Texts in Mathematics*. Springer, New York, NY, USA; 1985:xi+184.View ArticleGoogle Scholar - Jeffrey A:
*Handbook of Mathematical Formulas and Integrals*. 3rd edition. Elsevier Academic Press, San Diego, Calif, USA; 2004:xxvi+453.MATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.